Explicit block encodings of boundary value problems for many-body elliptic operators

Tyler Kharazi; Ahmad M. Alkadri; Jin-Peng Liu; Kranthi K. Mandadapu; K. Birgitta Whaley

doi:10.22331/q-2025-06-04-1764

Explicit block encodings of boundary value problems for many-body elliptic operators

Tyler Kharazi^1,2, Ahmad M. Alkadri³, Jin-Peng Liu^4,5,6, Kranthi K. Mandadapu^3,7, and K. Birgitta Whaley^1,2

¹Department of Chemistry, University of California, Berkeley
²Berkeley Quantum Information and Computation Center, University of California, Berkeley
³Department of Chemical and Biomolecular Engineering, University of California, Berkeley
⁴Yau Mathematical Sciences Center, Tsinghua University
⁵Center for Theoretical Physics, Massachusetts Institute of Technology
⁶Simons Institute and Department of Mathematics, University of California, Berkeley
⁷Chemical Sciences Division, Lawrence Berkeley National Laboratory, Berkeley

Published:	2025-06-04, volume 9, page 1764
Editor:	Di Fang
Eprint:	arXiv:2407.18347v4
Doi:	https://doi.org/10.22331/q-2025-06-04-1764
Citation:	Quantum 9, 1764 (2025).

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Abstract

Simulation of physical systems is one of the most promising use cases of future digital quantum computers. In this work we systematically analyze the quantum circuit complexities of block encoding the discretized elliptic operators that arise extensively in numerical simulations for partial differential equations, including high-dimensional instances for many-body simulations. When restricted to rectangular domains with separable boundary conditions, we provide explicit circuits to block encode the many-body Laplacian with separable periodic, Dirichlet, Neumann, and Robin boundary conditions, using standard discretization techniques from low-order finite difference methods. To obtain high-precision, we introduce a scheme based on periodic extensions to solve Dirichlet and Neumann boundary value problems using a high-order finite difference method, with only a constant increase in total circuit depth and subnormalization factor. We then present a scheme to implement block encodings of differential operators acting on more arbitrary domains, inspired by Cartesian immersed boundary methods. We then block encode the many-body convective operator, which describes interacting particles experiencing a force generated by a pair-wise potential given as an inverse power law of the interparticle distance. This work provides concrete recipes that are readily translated into quantum circuits, with depth logarithmic in the total Hilbert space dimension, that block encode operators arising broadly in applications involving the quantum simulation of quantum and classical many-body mechanics.

Featured image: Solutions of the Poisson equation with Dirichlet boundary conditions using the projection method established in this work, showing its ability to find solutions on domains that cannot be expressed as simple tensor products of one-dimensional domains. We provide explicit circuit constructions for these and more general kinds of boundary value problems that can be readily employed for cost estimation and algorithm analysis for applications in quantum computational solutions of classical partial differential equations in a broad range of settings.

Popular summary

High-dimensional partial differential equations (PDEs) arise when studying physical processes in terms of molecular constituents, with the number of dimensions depending on the microscopic variables of interest. Parabolic PDEs often describe the time evolution of a system under the influence of elliptic operators, a class of operators generalizing the Laplace operator. In many-body settings, the exponential growth in state space leads to an exponential increase in the complexity of direct classical solvers with particle number, known as the “curse of dimensionality”. Quantum computation offers an attractive alternative to solving these important, but classically intractable, high-dimensional problems.
Here, we analyze the quantum circuit complexities of block encoding the discretized elliptic operators that arise extensively in numerical simulations for classical PDEs, including high-dimensional instances for many-body simulations. We construct quantum circuit primitives and resource estimations to produce block encodings that encode discretized elliptic operators in a variety of boundary conditions. We further introduce techniques that implement these operators using periodic extensions to obtain high-precision approximations, as well as techniques for implementing them on irregular domains. We then present a scheme to efficiently block encode the many-body convective operator for particles interacting by a pair-wise potential, opening a route to efficient quantum algorithms to simulate high-dimensional Fokker-Planck and Smoluchowski-type equations for many-body systems.

► BibTeX data

@article{Kharazi2025explicitblock,
  doi = {10.22331/q-2025-06-04-1764},
  url = {https://doi.org/10.22331/q-2025-06-04-1764},
  title = {Explicit block encodings of boundary value problems for many-body elliptic operators},
  author = {Kharazi, Tyler and Alkadri, Ahmad M. and Liu, Jin-Peng and Mandadapu, Kranthi K. and Whaley, K. Birgitta},
  journal = {{Quantum}},
  issn = {2521-327X},
  publisher = {{Verein zur F{\"{o}}rderung des Open Access Publizierens in den Quantenwissenschaften}},
  volume = {9},
  pages = {1764},
  month = jun,
  year = {2025}
}

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[5] Michael Lubasch, Yuta Kikuchi, Lewis Wright, and Conor Mc Keever, "Quantum circuits for partial differential equations in Fourier space", Physical Review Research 7 4, 043326 (2025).

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